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Controlled concentration points and groups of divergence type. (English) Zbl 0872.22007
Johannson, Klaus (ed.), Low-dimensional topology. Proceedings of a conference, held May 18-26, 1992 at the University of Tennessee, Knoxville, TN, USA. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 3, 41-45 (1994).
For a nonelementary group $$\Gamma$$ of hyperbolic isometries acting on $$B^{d+1}$$, the critical exponent $$\delta(\Gamma)$$ is defined as the infimum of those $$\alpha$$ for which the series $$\Sigma_{\gamma\in\Gamma} e^{-\alpha(0,\gamma(0))}$$ converges, where $$(0,\gamma(0))$$ is the hyperbolic distance from $$0$$ to $$\gamma(0)$$. The group is said to be of convergence or divergence type according as the series above converges or diverges for $$\alpha=\delta(\Gamma)$$. A limit point $$p\in S^d$$ of $$\Gamma$$ is called a controlled concentration point if it has a neighbourhood $$U$$ such that for any connected neighbourhood $$V$$ of $$p$$ there exists an element $$\gamma\in\Gamma$$ such that $$p\in\gamma(V)$$ and $$\gamma(U)\subset V$$. In the paper under review the author uses results of his thesis to show that for a nonelementary group of divergence type the set of controlled concentration points has full Patterson-Sullivan measure in the set of limit points of $$\Gamma$$. In fact, the set of limit points satisfying a stronger condition and which he calls Myrberg-Agard concentration points, has full Patterson-Sullivan measure. This follows from results of the author obtained using S. Agard’s approach [Acta Math. 151, 231-252 (1983; Zbl 0532.30038)] and properties of the Sullivan-Patterson measure.
For the entire collection see [Zbl 0816.00026].

##### MSC:
 22E40 Discrete subgroups of Lie groups 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)